Solve problems involving satellite motion

Satellite Motion

A satellite is an object in space that orbits or revolves around another object. A natural satellite is an astronomical body that orbits a planet, like the moon. An artificial satellite is a manufactured object that orbits the earth, such as a satellite used for communication or weather monitoring.
Kepler’s Laws of Planetary Motion, are three scientific laws describing the motion of planets around the sun, but these are also applicable to satellites revolving around planets.
The First Law, also known as the Law of Orbits, states that all planets move in elliptical orbits, with the sun at one of the foci of the ellipse.
The Second Law, also known as the Law of Equal Areas, states that an imaginary line drawn from the sun to a planet sweeps out equal areas in equal intervals of time.
The Third Law, also known as the Law of Periods, states that the square of the period of a planet’s orbit is proportional to the cube of the semi-major axis of its orbit.

The gravitational force is the force of attraction between two objects with mass. In regards to satellite motion, the gravitational force pulls the satellite towards the planet.
Newton’s Law of Universal Gravitation describes the gravitational attraction as inversely proportional to the square of the distance between the objects and directly proportional to the product of their masses.
When a satellite is in orbit, the gravitational force is providing the necessary centripetal force to keep the satellite moving in a circular path. Hence, the formula for gravitational attraction (F = G(Mm/r^2)) can be equated to the formula for centripetal force (F = mv^2/r).

The orbital velocity of a satellite is the minimum velocity an object must have in order to remain in a stable orbit around a planet.
The orbital velocity can be determined by the formula v = √(GM/r), where v is the orbital velocity, G is the universal gravitational constant, M is the mass of the planet, and r is the radius of the orbit.
The orbital period of a satellite is the time it takes for a satellite to complete one full orbit around a planet. This can be found using the formula T = 2π√(r^3/GM).
Note that if a satellite is in a geostationary orbit, it has an orbital period equal to the rotational period of the planet, and as a result, seems to remain stationary when observed from the planet’s surface.

The total mechanical energy of a satellite in orbit is the sum of its kinetic and potential energies (E = K + U).
The kinetic energy (K) of a satellite is given by K = 1/2mv^2, where m is the mass of the satellite and v is its velocity.
The gravitational potential energy (U) at a given point in space due to a planet is given by U = -GMm/r, where M is the mass of the planet and r is the distance from the centre of the planet. Note the negative sign indicates the potential energy decreases as the distance increases.
The total mechanical energy of a satellite in orbit is always negative, indicating the satellite is bound to the planet — it does not have enough energy to escape the planet’s gravity.

Perturbations in a satellite’s orbit can be caused by several factors including non-uniformity in the gravitational field of the planet, gravitational effects from the sun, moon or other bodies, atmospheric drag, and pressure from solar radiation.
Monitoring stations on earth or in space track satellites and measure their position and velocity. These measurements, along with knowledge about the forces that can affect a satellite’s motion, may be used to predict future positions and velocities (orbit determination). And if needed, course corrections can be made using momentum wheels or thrusting mechanisms onboard the satellite.